C -Algebras and Finite-Dimensional Approximations This page intentionally left blank C*-Algebra s an d Finite-Dimensiona l Approximation s Nathania l P . Brow n Narutak a Ozaw a Graduate Studies in Mathematics Volume 88 American Mathematical Society Providence, Rhode Island Editorial Board David Cox (Chair) Walter Craig N. V. Ivanov Steven G. Krantz 2000 Mathematics Subject Classification. Primary 46L05, 46L06, 46L07, 46L10, 46L35, 46L55, 05C25, 22D25, 43A07. For additional information and updates on this book, visit www.ams.org/bookpages/gsm-88 Library of Congress Cataloging-in-Publication Data Brown, Nathanial P. (Nathanial Patrick), 1972- C*-algebras and finite-dimensional approximations / Nathanial P. Brown, Narutaka Ozawa. p. cm. — (Graduate studies in mathematics, ISSN 1065-7339 ; v. 88) Includes bibliographical references and index. ISBN-13: 978-0-8218-4381-9 (alk. paper) ISBN-10: 0-8218-4381-8 (alk. paper) 1. C*-algebras. 2. Finite groups. I. Ozawa, Narutaka, 1974- II. Title. QA326 .B758 2008 512/.556—dc22 2007060566 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for such permission should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island 02904-2294, USA. Requests can also be made by e-mail to [email protected]. © 2008 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. @ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http: //www. ams. org/ 10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 08 Contents Preface xi Chapter 1. Fundamental Facts 1 §1.1. Notation 1 §1.2. C*-algebras 2 §1.3. Von Neumann algebras 3 §1.4. Double duals 5 §1.5. Completely positive maps 9 §1.6. Arveson's Extension Theorem 17 §1.7. Voiculescu's Theorem 18 Part 1. Basic Theory Chapter 2. Nuclear and Exact C*-Algebras: Definitions, Basic Facts and Examples 25 §2.1. Nuclear maps 25 §2.2. Nonunital technicalities 28 §2.3. Nuclear and exact C*-algebras 32 §2.4. First examples 38 §2.5. C*-algebras associated to discrete groups 42 §2.6. Amenable groups 48 §2.7. Type I C*-algebras 55 §2.8. References 58 Chapter 3. Tensor Products 59 VI Contents §3.1. Algebraic tensor products 59 §3.2. Analytic preliminaries 66 §3.3. The spatial and maximal C*-norms 72 §3.4. Takesaki's Theorem 77 §3.5. Continuity of tensor product maps 82 §3.6. Inclusions and The Trick 85 §3.7. Exact sequences 92 §3.8. Nuclearity and tensor products 99 §3.9. Exactness and tensor products 105 §3.10. References 112 Chapter 4. Constructions 115 §4.1. Crossed products 115 §4.2. Integer actions 121 §4.3. Amenable actions 124 §4.4. X xi T-algebras 129 §4.5. Compact group actions and graph C*-algebras 133 §4.6. Cuntz-Pimsner algebras 136 §4.7. Reduced amalgamated free products 154 §4.8. Maps on reduced amalgamated free products 157 §4.9. References 165 Chapter 5. Exact Groups and Related Topics 167 §5.1. Exact groups 167 §5.2. Groups acting on trees 176 §5.3. Hyperbolic groups 182 §5.4. Subgroups of Lie groups 193 §5.5. Coarse metric spaces 194 §5.6. Groupoids 200 §5.7. References 209 Chapter 6. Amenable Traces and Kirchberg's Factorization Property 211 §6.1. Traces and the right regular representation 211 §6.2. Amenable traces 214 §6.3. Some motivation and examples 223 §6.4. The factorization property and Kazhdan's property (T) 227 References 235 Contents vn Chapter 7. Quasidiagonal C*-Algebras 237 §7.1. The definition, easy examples and obstructions 237 §7.2. The representation theorem 243 §7.3. Homotopy invariance 247 §7.4. Two more examples 252 §7.5. External approximation 255 §7.6. References 260 Chapter 8. AF Embeddability 261 §8.1. Stable uniqueness and asymptotically commuting diagrams 261 §8.2. Cones over exact RFD algebras 267 §8.3. Cones over general exact algebras 268 §8.4. Homotopy invariance 274 §8.5. A survey 279 §8.6. References 282 Chapter 9. Local Reflexivity and Other Tensor Product Conditions 283 §9.1. Local reflexivity 284 §9.2. Tensor product properties 285 §9.3. Equivalence of exactness and property C 293 §9.4. Corollaries 297 §9.5. References 299 Chapter 10. Summary and Open Problems 301 §10.1. Nuclear C*-algebras 301 §10.2. Exact C*-algebras 303 §10.3. Quasidiagonal C*-algebras 306 §10.4. Open problems 309 Part 2. Special Topics Chapter 11. Simple C*-Algebras 313 §11.1. Generalized inductive limits 313 §11.2. NF and strong NF algebras 317 §11.3. Inner quasidiagonality 323 §11.4. Excision and Popa's technique 328 §11.5. Connes's uniqueness theorem 335 §11.6. References 337 viii Contents Chapter 12. Approximation Properties for Groups 339 §12.1. Kazhdan's property (T) 339 §12.2. The Haagerup property 354 §12.3. Weak amenability 361 §12.4. Another approximation property 369 §12.5. References 374 Chapter 13. Weak Expectation Property and Local Lifting Property 375 §13.1. The local lifting property 375 §13.2. Tensorial characterizations of the LLP and WEP 378 §13.3. The QWEP conjecture 380 §13.4. Nonsemisplit extensions 385 §13.5. Norms on 1ffi(£2) 81ffi(£2) 388 §13.6. References 391 Chapter 14. Weakly Exact von Neumann Algebras 393 §14.1. Definition and examples 393 §14.2. Characterization of weak exactness 397 §14.3. References 403 Part 3. Applications Chapter 15. Classification of Group von Neumann Algebras 407 §15.1. Subalgebras with noninjective relative commutants 407 §15.2. On bi-exactness 411 §15.3. Examples 414 §15.4. References 420 Chapter 16. Herrero's Approximation Problem 421 §16.1. Description of the problem 421 §16.2. C* -preliminaries 423 §16.3. Resolution of Herrero's problem 425 §16.4. Counterexamples 426 §16.5. References 429 Chapter 17. Counterexamples in K-Homology and K-Theory 431 §17.1. BDF preliminaries 431 §17.2. Property (T) and Kazhdan projections 435 §17.3. Ext need not be a group 438 Contents IX §17.4. Topology on Ext 439 §17.5. References 441 Part 4. Appendices Appendix A. Ultrafilters and Ultraproducts 445 Appendix B. Operator Spaces, Completely Bounded Maps and Duality 449 Appendix C. Lifting Theorems 459 Appendix D. Positive Definite Functions, Cocycles and Schoenberg's Theorem 463 Appendix E. Groups and Graphs 471 Appendix F. Bimodules over von Neumann Algebras 479 Bibliography 493 Notation Index 503 Subject Index 505 This page intentionally left blank Preface This is a book about C*-algebras, various types of approximation, and a few of the surprising applications that have been recently discovered. In short, we will study approximation theory in the context of operator algebras. Approximation is ubiquitous in mathematics; when the object of inter• est cannot be studied directly, we approximate by tractable relatives and pass to a limit. In our context this is particularly important because C*- algebras are (almost always) infinite dimensional and we can say precious little without the help of approximation theory. Moreover, most concrete examples enjoy some sort of finite-dimensional localization; hence it is very important to exploit these features to the fullest. Indeed, over the years approximation theory has been at the heart of many of the deepest, most important results: Murray and von Neumann's uniqueness theorem for the hyperfinite Ili-factor and Connes's remarkable extension to the injective realm; Haagerup's discovery that reduced free group C*-algebras have the metric approximation property; Higson and Kasparov's resolution of the Baum-Connes conjecture for Haagerup's groups; Popa's work on subfactors and Cartan subalgebras; Voiculescu's whole free entropy industry, which is defined via approximation; Elliott's classification program, which collapses without approximate intertwining arguments; and one can't forget the influ• ential work of Choi, Effros, and Kirchberg on nuclear and exact C*-algebras. Approximation is everywhere; it is powerful, important, the backbone of countless breakthroughs. We intend to celebrate it. This subject is a func• tional analyst's delight, a beautiful mixture of hard and soft analysis, pure joy for the technically inclined. Our wheat may be other texts' chaff, but we see no reason to hide our infatuation with the grace and power which is ap• proximation theory. We don't mean to suggest that mastering technicalities XI Xll Preface is the point of operator algebras - it isn't. We simply hope to elevate them from a necessary ally to a revered friend. Also, one shouldn't think these pages are a one-stop shopping place for all aspects of approximation the• ory - they aren't. The main focus is nuclearity and exactness, with several related concepts and a few applications thrown in for good measure. From the outset of this project, we were torn between writing user- friendly notes which students would appreciate - many papers in this subject are notoriously difficult to read - or sticking to an expert-oriented, research- monograph level of exposition. In the end, we decided to split the difference. Part 1 of these notes is written with the beginner in mind, someone who just finished a first course in operator algebras (C*- and W*-algebras).